I wrote a detailed explanation of this here that may be helpful.
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MJDApr 24 '14 at 12:58

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The equivalence class $[x]$ is the set of all elements (in $X$) that are equivalent to $x$; by the equivalence relation $\sim$.
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Graham KempApr 24 '14 at 13:06

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"I have struggled as to what this actually mean" -- do you mean that you don't understand the notation, or do you mean that you don't understand the significance of equivalence classes? If the former then there's kind of no point someone just telling you what equivalence classes are, you also need to learn to read the notation for yourself. Suppose I ask you to tell me the equivalence classes of the relation "ends with the same decimal digit" on the set of integers from 1 to 100, would you know how to do that?
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Steve JessopApr 24 '14 at 18:38

4 Answers
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Equivalence classes are sets of elements which are all equivalent between them. For instance if the equivalence relation $\sim$ is "having the same sex", then there are two equivalence classes in the world: boys and girls (if we forget the ambiguous cases).
In the same way, if the equivalence relation is "being born the same year", then each year yields a different equivalence class of all the people from this year.

To sum up, an equivalence relation cuts the universe into "potatoes" of elements: inside a potato, all elements are equivalent to each other, and a potato is called an equivalence class.

The set $[x]$ consists, as you write, exactly of those elements $y$ in $X$ satisfying that $x\sim y$. That is, given the $x$, the equivalence class is just a subset of $X$. Note in particular, that if $x \sim y$, then $[x] = [y]$.

Consider the example: $X = \mathbb{Z}$. Define the equivalence relation that $x\sim y$ if and only if $2 \mid x- y$. That is $x$ and $y$ are in the same equivalence class exactly when their difference is divisible by $2$.

So what is $[0]$? It is not hard to see that this is just all the element in $\mathbb{Z}$ that are divisible by $2$: $y \sim 0$ means $2 \mid y - 0$.

Bonus question: How many equivalence classes do you have? Hint: $2$

Extra bonus question: Write down the two equivalence classes. Answer: If you think about it, it is not hard to see that the $2$ equivalence classes are exactly the odd and even numbers.

Let the ground set be people, and say that two people are equivalent if they have the same type of sex organs. Then, every man is equivalent to every other man. The set of all men is an equivalence class, as is the set of all women.

Let the ground set be integers, and say that two integers are equivalent if their difference is a multiple of 2. Every even integer is equivalent to every other even integer (since the difference of two evens is even). Every odd integer is equivalent to every other odd integer similarly. Hence the two equivalence classes are the set of all even integers, and the set of all odd integers.